pargpaRC {lmomco} | R Documentation

## Estimate the Parameters of the Generalized Pareto Distribution with Right-Tail Censoring

### Description

This function estimates the parameters (xi, α, and kappa) of the Generalized Pareto distribution given the “B”-type L-moments (through the B-type Probability-Weighted Moments) of the data under right censoring conditions (see `pwmRC`). The “B”-type L-moments in terms of the parameters are

λ^B_1 = xi + α m_1 mbox{,}

λ^B_2 = α (m_1 - m_2) mbox{,}

λ^B_3 = α (m_1 - 3m_2 + 2m_3)mbox{,}

λ^B_4 = α (m_1 - 6m_2 + 10m_3 - 5m_4)mbox{, and}

λ^B_5 = α (m_1 - 10m_2 + 30m_3 - 35m_4 + 14m_5)mbox{,}

noindent where m_r = lbrace 1-(1-zeta)^{r+kappa} rbrace/(r+kappa) and zeta is the right-tail censor fraction or the probability mathrm{Pr}lbrace rbrace that x is less than the quantile at zeta nonexceedance probability: (mathrm{Pr}lbrace x < X(zeta) rbrace). Finally, the `RC` in the function name is to denote `R`ight-tail `C`ensoring.

### Usage

    pargpaRC(lmom,zeta=1,lower=-1,upper=20,checklmom=TRUE)

### Arguments

`lmom` | A B-type L-moment object created by a function such as `pwm2lmom` from B-type Probability-Weighted Moments from `pwmRC`.
`zeta` | The censoring fraction. The number of samples observed (noncensored) divided by the total number of samples.
`lower` | The lower value for kappa for a call to the `optimize` function. For the L-moments of the distribution to be valid kappa > -1.
`upper` | The upper value for kappa for a call to the `optimize` function. Hopefully, a large enough default is chosen for real-world data sets.
`checklmom` | Should the `lmom` be checked for validity using the `are.lmom.valid` function. Normally this should be left as the default and it is very unlikely that the L-moments will not be viable (particularly in the tau_4 and tau_3 inequality). However, for some circumstances or large simulation exercises then one might want to bypass this check.

### Details

The `optimize` function is used to numerically solve for the shape parameter kappa. No test or evaluation is made on the quality of the minimization. Users should concults the contents of the `optim` portion of the returned list. Finally, this function should return the same parameters if zeta=1 as the `pargpa` function.

### Value

An R `list` is returned.

`type` | The type of distribution: `gpa`.
`para` | The parameters of the distribution.
`zeta` | The right-tail censoring fraction.
`source` | The source of the parameters: “pargpaRC”.
`optim` | The `list` returned by the `optimize` function.

### Author(s)

W.H. Asquith

### References

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, vol. 52, p. 105–124.

Hosking, J.R.M., 1995, The use of L-moments in the analysis of censored data, in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan, chapter 29, CRC Press, Boca Raton, Fla., pp. 546–560.

### See Also

`lmomgpa`, `lmomgpaRC`, `pargpa`, `cdfgpa`, `quagpa`

### Examples

    n         <- 60 # samplesize
    para      <- vec2par(c(1500,160,.3),type="gpa") # build a GPA parameter set
    fakedata  <- quagpa(runif(n),para) # generate n simulated values
    threshold <- 1700 # a threshold to apply the simulated censoring
    fakedata  <- sapply(fakedata,function(x) { if(x > threshold)
                                               return(threshold) else return(x)
                                             })
    lmr       <- lmoms(fakedata) # Ordinary L-moments without considering
                                 # that the data is censored
    estpara   <- pargpa(lmr) # Estimated parameters of parent

    pwm2     <- pwmRC(fakedata,threshold=threshold) # compute censored PWMs
    typeBpwm <- pwm2$Bbetas # the B-type PWMs
    zeta     <- pwm2$zeta # the censoring fraction

    cenpara <- pargpaRC(pwm2lmom(typeBpwm),zeta=zeta) # Estimated parameters
    F       <- nonexceeds() # nonexceedance probabilities for plotting purposes

    # Visualize some data
    plot( F,quagpa(F,para),type='l', lwd=3) # The true distribution
    lines( F,quagpa(F,estpara),col=3) # Green estimated in the ordinary fashion
    lines(F,quagpa(F,cenpara),col=2) # Red, consider that the data is censored
    # now add in what the drawn sample looks like.
    PP <- pp(fakedata) # plotting positions of the data
    points(PP,sort(fakedata)) # sorting is needed!
    # Interpretation. You should see that the red line more closely matches
    # the heavy black line. The green line should be deflected to the right
    # and pass through the values equal to the threshold, which reflects the
    # much smaller L-skew of the ordinary L-moments compared to the type-B
    # L-moments.

    # Assertion, given some PWMs or L-moments, if zeta=1 then the parameter
    # estimates must be identifical. The following provides a demonstration.
    para1 <- pargpaRC(pwm2lmom(typeBpwm),zeta=1)
    para2 <- pargpa(pwm2lmom(typeBpwm))
    str(para1)
    str(para2)

[Package lmomco version 0.92 Index]